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A368152 Triangular array T(n,k), read by rows: coefficients of strong divisibility sequence of polynomials p(1,x) = 1, p(2,x) = 1 + 3*x, p(n,x) = u*p(n-1,x) + v*p(n-2,x) for n >= 3, where u = p(2,x), v = 3 - x^2.

%I A368152 #14 Jan 22 2024 06:03:04
%S A368152 1,1,3,4,6,8,7,27,25,21,19,66,126,90,55,40,204,392,504,300,144,97,522,
%T A368152 1363,1884,1851,954,377,217,1425,4065,7281,8011,6435,2939,987,508,
%U A368152 3642,12332,24606,34044,31446,21524,8850,2584,1159,9441,35236,82020,127830
%N A368152 Triangular array T(n,k), read by rows: coefficients of strong divisibility sequence of polynomials p(1,x) = 1, p(2,x) = 1 + 3*x, p(n,x) = u*p(n-1,x) + v*p(n-2,x) for n >= 3, where u = p(2,x), v = 3 - x^2.
%C A368152 Because (p(n,x)) is a strong divisibility sequence, for each integer k, the sequence (p(n,k)) is a strong divisibility sequence of integers.
%H A368152 Rigoberto Flórez, Robinson Higuita, and Antara Mukherjee, <a href="http://math.colgate.edu/~integers/s14/s14.Abstract.html">Characterization of the strong divisibility property for generalized Fibonacci polynomials</a>, Integers, 18 (2018), 1-28, Paper No. A14.
%F A368152 p(n,x) = u*p(n-1,x) + v*p(n-2,x) for n >= 3, where p(1,x) = 1, p(2,x) = 1 + 3*x, u = p(2,x), and v = 3 - x^2.
%F A368152 p(n,x) = k*(b^n - c^n), where k = -1/sqrt(13 + 6*x + 5*x^2), b = (1/2)*(3*x + 1 - 1/k), c = (1/2)*(3*x + 1 + 1/k).
%e A368152 First eight rows:
%e A368152     1
%e A368152     1    3
%e A368152     4    6    8
%e A368152     7   27   25   21
%e A368152    19   66  126   90   55
%e A368152    40  204  392  504  300  144
%e A368152    97  522 1363 1884 1851  954  377
%e A368152   217 1425 4065 7281 8011 6435 2939 987
%e A368152 Row 4 represents the polynomial p(4,x) = 7 + 27*x + 25*x^2 + 21*x^3, so (T(4,k)) = (7,27,25,21), k=0..3.
%t A368152 p[1, x_] := 1; p[2, x_] := 1 + 3 x; u[x_] := p[2, x]; v[x_] := 3 - x^2;
%t A368152 p[n_, x_] := Expand[u[x]*p[n - 1, x] + v[x]*p[n - 2, x]]
%t A368152 Grid[Table[CoefficientList[p[n, x], x], {n, 1, 10}]]
%t A368152 Flatten[Table[CoefficientList[p[n, x], x], {n, 1, 10}]]
%Y A368152 Cf. A006130 (column 1); A001906 (p(n,n-1)); A090017 (row sums), (p(n,1)); A002605 (alternating row sums), (p(n,-1)); A004187, (p(n,2)); A004254, (p(n,-2)); A190988, (p(n,3)); A190978 (unsigned), (p(n,-3)); A094440, A367208, A367209, A367210, A367211, A367297, A367298, A367299, A367300, A367301, A368150, A368151.
%K A368152 nonn,tabl
%O A368152 1,3
%A A368152 _Clark Kimberling_, Jan 20 2024